Nonnegative rank of a matrix with one negative eigenvalue

We show that a rank-three symmetric matrix with exactly one negative eigenvalue can have arbitrarily large nonnegative rank.     

On perturbations of matrix pencils with real spectra. II

A well-known result on spectral variation of a Hermitian matrix due to Mirsky is the following: Let and be two Hermitian matrices, and let and be their eigenvalues arranged in ascending order. Then for any unitarily invariant norm . In this paper, we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra. The much studied case of definite pencils is included in this.

     

Efficient eigenvalue computation for quasiseparable Hermitian matrices under low rank perturbations

In this paper we address the problem of efficiently computing all the eigenvalues of a large N×N Hermitian matrix modified by a possibly non Hermitian perturbation of low rank. Previously proposed fast adaptations of the QR algorithm are considerably simplified by performing a preliminary transformation of the matrix by similarity into an upper Hessenberg form. The transformed matrix can be specified by a small set of parameters which are easily updated during the QR process. The resulting structured QR iteration can be carried out in linear time using linear memory storage. Moreover, it is proved to be backward stable. Numerical experiments show that the novel algorithm outperforms available implementations of the Hessenberg QR algorithm already for small values of N.      

On perturbations of matrix pencils with real spectra, a revisit

This paper continues earlier studies by Bhatia and Li on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectra. A unifying framework for creating crucial perturbation equations is developed. With the help of a recent result on generalized commutators involving unitary matrices, new and much sharper bounds are obtained.

     

Smooth rank one perturbations of selfadjoint operators

Let be a selfadjoint operator in a Hilbert space with inner product . The rank one perturbations of have the form , , for some element . In this paper we consider smooth perturbations, i.e. we consider for some . Function-theoretic properties of their so-called -functions and operator-theoretic consequences will be studied.

     

Singular continuous spectrum under rank one perturbations and localization for random hamiltonians

We consider a selfadjoint operator, A, and a selfadjoint rank-one projection, P, onto a vector, φ, which is cyclic for A. In terms of the spectral measure dμ^A_φ, we give necessary and sufficient conditions for A + λ P to have empty singular continuous spectrum or to have only point spectrum for a.e. λ. We apply these results to questions of localization in the one- and multi-dimensional Anderson models.     

Eigenvalues of rank one perturbations of unstructured matrices

Let A be a fixed complex matrix and let $u\text{,}v$ be two vectors. The eigenvalues of matrices $A+\tau {\mathit{uv}}^{?}$ $(\tau \in \mathbb{R})$ form a system of intersecting curves. The dependence of the intersections on the vectors $u\text{,}v$ is studied.     

The largest eigenvalue of small rank perturbations of Hermitian random matrices

We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. We consider random Hermitian matrices with independent Gaussian entries M _ij ,i≤j with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large N limit, various limiting distributions depending on both the eigenvalues of the matrix and its rank. This rank is also allowed to increase with N in some restricted way. An erratum to this article is available at .     

The threshold effects for a family of Friedrichs models under rank one perturbations

A family of Friedrichs models under rank one perturbations h_μ(p), p(−π,π]³, μ>0, associated to a system of two particles on the three-dimensional lattice is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of h_μ(p) for all non-trivial values of p under the assumption that h_μ(0) has either a threshold energy resonance (virtual level) or a threshold eigenvalue. The threshold energy expansion for the Fredholm determinant associated to a family of Friedrichs models is also obtained.     

On the rank one perturbations of the Heisenberg commutation relation and unbounded subnormal operators

In this paper, a functional model of rank one perturbation of the Heisenberg commutation relation is established. In some cases, it turns out to be unbounded subnormal.     

Low rank matrix recovery from rank one measurements

We study the recovery of Hermitian low rank matrices $X\in {\mathbb{C}}^{n\times n}$ from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form $a_j{a}_j^{?}$ for some measurement vectors $a_1,\dots ,a_m$, i.e., the measurements are given by $b_j=\mathrm{tr}(X{a}_j{a}_j^{?})$. The case where the matrix $X=x{x}^{?}$ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements $b_j=|〈x,a_j〉{|}^2$) via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors $a_j$, $j=1,\dots ,m$, being chosen independently at random according to a standard Gaussian distribution, or $a_j$ being sampled independently from an (approximate) complex projective t-design with $t=4$. In the Gaussian case, we require $m\ge Crn$ measurements, while in the case of 4-designs we need $m\ge \mathit{Cr}n\log ?(n)$. Our results are uniform in the sense that one random choice of the measurement vectors $a_j$ guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Krahmer and Kueng. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.     

Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations

udy the perturbation theory of structured matrices under structured rank one perturbations, and then focus on several classes of complex matrices. Generic Jordan structures of perturbed matrices are identified. It is shown that the perturbation behavior of the Jordan structures in the case of singular J-Hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in [18, 28, 30, 31]. Thus a generic structured perturbation would not be generic if considered as an unstructured perturbation. In other settings of structured matrices, the generic perturbation behavior of the Jordan structures, within the confines imposed by the structure, follows the pattern of that of unstructured perturbations.     

Perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations

For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in the sense of algebraic geometry. Special attention is paid to the perturbation behavior of the sign characteristic. Typically, under such a perturbation, for every given eigenvalue, the largest Jordan block of the eigenvalue is destroyed and (in case the eigenvalue is real) all other Jordan blocks keep their sign characteristic. The new eigenvalues, i.e. those eigenvalues of the perturbed matrix that are not eigenvalues of the original matrix, are typically simple, and in some cases information is provided about their sign characteristic (if the new eigenvalue is real). The main results are proved by using the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky canonical form and on general results concerning rank one perturbations obtained.     

On spectral variations under bounded real matrix perturbations

Summary In this paper we investigate the set of eigenvalues of a perturbed matrix {ie509-1} whereA is given and ^{n × n}, ||< is arbitrary. We determine a lower bound for thisspectral value set which is exact for normal matricesA with well separated eigenvalues. We also investigate the behaviour of the spectral value set under similarity transformations. The results are then applied tostability radii which measure the distance of a matrixA from the set of matrices having at least one eigenvalue in a given closed instability domain _b.     

On finite rank perturbations of definitizable operators

It was shown by P. Jonas and H. Langer that a selfadjoint definitizable operator A in a Krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator B is selfadjoint and the resolvent set ρ(B) is nonempty. It is the aim of this note to prove a more general variant of this perturbation result where the assumption on ρ(B) is dropped. As an application a class of singular ordinary differential operators with indefinite weight functions is studied.     

Point spectrum and mixed spectral types for rank one perturbations

We consider examples of rank one perturbations with a cyclic vector for . We prove that for any bounded measurable set , an interval, there exist so that
eigenvalue agrees with up to sets of Lebesgue measure zero. We also show that there exist examples where has a.c. spectrum for all , and for sets of 's of positive Lebesgue measure, also has point spectrum in , and for a set of 's of positive Lebesgue measure, also has singular continuous spectrum in .

     

Upper eigenvalue bounds for pencils of matrices

Eigenvalue bounds are obtained for pencils of matrices A ? vB where A is a Stieltjes matrix and B is positive definite, under assumptions suitable for the estimation of asymptotic convergence rates of factorization iterative methods, where B represents the approximate factorization of A. The upper bounds obtained depend on the “connectivity” structure of the matrices involved, which enters through matrix graph considerations; in addition, a more classical argument is used to obtain a lower bound. Potential applications of these results include a partial confirmation of Gustafsson's conjecture concerning the nonnecessity of Axelsson's perturbations.     

Lower eigenvalue bounds for pencils of matrices

A procedure is set up for obtaining lower eigenvalue bounds for pencils of matrices A—vB where A is a Stieltjes matrix and B is positive definite, under assumptions suitable for the estimation of asymptotic convergence rates of locally perturbed factorization iterative schemes. Using these results and a formerly developed approach for estimating upper bounds, we widely confirm Gustafsson's conjecture concerning the nonnecessity of Axelsson's perturbations. In so doing, we however keep local perturbations, thereby enlarging the number of applications where their sufficiency is proven; their necessity remains, on the other hand, an open question.